Linear Algebra

$$\newcommand{\P}[]{\unicode{xB6}} \newcommand{\AA}[]{\unicode{x212B}} \newcommand{\empty}[]{\emptyset} \newcommand{\O}[]{\emptyset} \newcommand{\Alpha}[]{Α} \newcommand{\Beta}[]{Β} \newcommand{\Epsilon}[]{Ε} \newcommand{\Iota}[]{Ι} \newcommand{\Kappa}[]{Κ} \newcommand{\Rho}[]{Ρ} \newcommand{\Tau}[]{Τ} \newcommand{\Zeta}[]{Ζ} \newcommand{\Mu}[]{\unicode{x039C}} \newcommand{\Chi}[]{Χ} \newcommand{\Eta}[]{\unicode{x0397}} \newcommand{\Nu}[]{\unicode{x039D}} \newcommand{\Omicron}[]{\unicode{x039F}} \DeclareMathOperator{\sgn}{sgn} \def\oiint{\mathop{\vcenter{\mathchoice{\huge\unicode{x222F}\,}{\unicode{x222F}}{\unicode{x222F}}{\unicode{x222F}}}\,}\nolimits} \def\oiiint{\mathop{\vcenter{\mathchoice{\huge\unicode{x2230}\,}{\unicode{x2230}}{\unicode{x2230}}{\unicode{x2230}}}\,}\nolimits}$$

Linear Algebra

Definiteness

A square matrix $$\displaystyle A$$ is positive definite if for all vectors $$\displaystyle x$$, $$\displaystyle x^T A x \gt 0$$.
If the inequality is weak ($$\displaystyle \geq$$) then the matrix is positive semi-definite.

Properties

• If the determinant of every upper-left submatrix is positive then the matrix is positive-definite.
• If $$\displaystyle A$$ is PSD then $$\displaystyle A^T$$ is PSD.
• The sum of PSD matrices is PSD.

Examples

Examples of PSD matrices:

• The identity matrix is PSD
• $$\displaystyle x x^T$$ is PSD for any vector $$\displaystyle x$$